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MAE 4262: ROCKETS AND MISSION ANALYSIS
Single and Multi-Stage Rockets
September 9, 2014
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
1
SINGLE-STAGE SOUNDING ROCKET SUMMARY
• Want to reduce burn time as much as possible while accelerating against a gravity field
• Short burn time reduces energy consumed in lifting propellants
• Very short burn time implies very high accelerations
– Structural limitations
– High mass flows, lots of weight for nozzles, turbo-machinery, cooling, etc.
– Drag goes as V2
• Is there an optimum acceleration for a given rocket configuration?
• In limit of no drag and no gravity, burn time has no influence on velocity increment
1ln12
ln
2
1
1
ln
111ln
22
max
2
RR
Rtu
g
Ruh
gtR
Rtutuh
gtt
t
RutV
bee
bbebeb
be Velocity during
Powered flight
Height at burnout
Maximum altitude
2
PRELIMINARY DEFINITIONS
• Total mass of rocket, Mo, may be written as sum of 3 primary components:– Payload mass, ML
– Propellant mass, MP
– Structural mass, MS
• Includes everything but payload and propellant• Engines, tanks, controls, etc.
• If rocket consumes all its propellant during firing, burnout mass consists of structure and payload:
• NOTE: Other texts and references will breakdown rocket components in various ways and into many more parts (i.e., Sutton, Kerrebrock, Turner, Humble)
SPLo MMMM
SLb MMM
3
DEFINITIONS
Symbol Ratio Description
R SL
o
b
o
MM
M
M
MR
Mass Ratio: initial mass / mass at the end of the thrust period. Want this ratio large.
SP
L
Lo
L
MM
M
MM
M
Payload Ratio: ratio of payload to everything but payload. Want this large, but larger the payload, the lower maximum attainable velocity.
Lo
Lb
SP
S
MM
MM
MM
M
Structural Coefficient: ratio of the structural weight to everything but the payload. Want this small.
Lo
P
SP
P
MM
M
MM
M
Propellant Ratio: Ratio of propellant
to everything but the payload.
4
PRELIMINARY DEFINITIONS
• Using previous definitions, we can write mass ratio as:
• Also note that propellant ratio and structural coefficient are related by:
1
R
1
5
PICTURES OF DEFINITIONS
Payload
Propellant
Structure
Rocket InitialPropellant is Full
SPLo MMMM
Mo = + +
6
PICTURES OF DEFINITIONS
Payload
Propellant
Structure
Rocket FinalPropellant is EmptyMass at Burnout
SLb MMM
Mb = +
7
MASS RATIO DEFINITION
Payload
Propellant
Structure
SL
o
b
o
MM
M
M
MR
R=+ +
+
Roc
ket I
niti
alR
ocket Final
8
SUMMARY: SINGLE-STAGE ROCKETS
Payload
Propellant
Structure
Lo
P
SP
P
MM
M
MM
M
=
+
Lo
Lb
SP
S
MM
MM
MM
M
=+
SP
L
Lo
L
MM
M
MM
M
=
+
SL
o
b
o
MM
M
M
MR
R =
+ +
+
1
R 1 12
MULTISTAGE ROCKETS
• Main idea is to discard empty tanks and extra structure as rocket travels, so that this mass is not subjected to gravity losses
• Large engines used for initial high thrust phase, may produce excessive accelerations when propellant is nearly consumed
• Multistage rocket is a series of individual vehicles or stages, each with its own structure, tanks and engines
• Each stage accelerates payload before being detached
Two points:
1. Stages are ordered in number of firing
2. Analysis of multistage rockets is similar to that for single stage
– Payload for an particular stage is the mass of all subsequent stages
13
MULTISTAGE ROCKET EXAMPLE
1
ML
3
2
Total Mass 1: Mo1=MP1+MS1+Mo2
Total Mass 2: Mo2=MP2+MS2+Mo3
Total Mass 3: Mo3=MP3+MS3+ML
Total Mass i: Moi=MPi+MSi+Mo(i+1)
14
MULTISTAGE ROCKET EXAMPLE
1
ML
3
2
Total Mass 1: Mo1=MP1+MS1+Mo2
Payload for Stage 1: ML1=Mo2
Total Mass 2: Mo2=MP2+MS2+Mo3
Payload for Stage 2: ML2=Mo3
Total Mass 3: Mo3=MP3+MS3+ML
Payload for Stage 3: ML3=ML
Total Mass i: Moi=MPi+MSi+Mo(i+1)
Payload for Stage i: MLi=Mo(i+1) 15
PAYLOAD RATIO: MULTISTAGE ROCKETS
1
ML
21
2
)1(
)1(
oo
o
iooi
io
Lo
Li MM
M
MM
M
MM
M
The payload ratio for stage 1 is:
1
16
PAYLOAD RATIO: MULTISTAGE ROCKETS
ML
32
3
)1(
)1(
oo
o
iooi
io
Lo
Li MM
M
MM
M
MM
M
The payload ratio for stage 2 is:
2
2
17
PAYLOAD RATIO: MULTISTAGE ROCKETS
ML
Lo
L
iooi
io
Lo
Li MM
M
MM
M
MM
M
3)1(
)1(
The payload ratio for stage 3 is:
3
3
18
STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS
1
ML
The structural coefficient for stage 1 is:
1
21
1
)1( oo
S
iooi
Si
SP
Si MM
M
MM
M
MM
M
19
STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS
ML
The structural coefficient for stage 2 is:
2
32
2
)1( oo
S
iooi
Si
SP
Si MM
M
MM
M
MM
M
2
20
STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS
ML
The structural coefficient for stage 3 is:
3
Lo
S
iooi
Si
SP
Si MM
M
MM
M
MM
M
3
3
)1(
3
21
SUMMARY: MULTISTAGE ROCKETS
)1(
)1(
iooi
ioi MM
M
)1(
iooi
sii MM
M
)1(
)1(
iooi
iobii MM
MM
bi
oii M
MR
ii
iiR
1
22
PROTON (SOVIET)• First Launch: July 1965 • Flight Rate: 13 per year• Capability: 44,100 lb to LEO; 12,100 lb to GTO; 4,850 lb to GEO
• Originally intended as a ballistic missile but converted to a space launch vehicle during development
• Two, three, and four-stage versions were developed • Used to launch satellites into GEO, interplanetary spacecraft, and manned
space stations such as Salyut and Mir
• Three or four-stage liquid-fueled vehicle – Stage 1 has six strap-on boosters with RD-253 engines burning N2O4
fed from the core stage 1 tank with UDMH fuel carried in the strap-on tanks, generating a total of 1,986,000 lb of thrust
– Stage 2 has four RD-0210 sustainer engines burning N2O4/UDMH fed from stage 2 tank, generating a total of 540,000 lb of thrust
– Stage 3 has one RD-473 engine with four verniers burning N2O4/UDMH, generating a total thrust of 142,000 lb
– Stage 4 has one RD-58 burning LO2/kerosene, generating a total thrust of 19,100 lb
• Length: 197 ft • Launch Weight: 1,550,000 lb • Diameter 22.6 ft • Liftoff Thrust: 1,986,000 lb • Payload Fairing: 24.6 ft x 12 ft 24